78,646 research outputs found
Non-parametric learning critical behavior in Ising partition functions: PCA entropy and intrinsic dimension
We provide and critically analyze a framework to learn critical behavior in
classical partition functions through the application of non-parametric methods
to data sets of thermal configurations. We illustrate our approach in phase
transitions in 2D and 3D Ising models. First, we extend previous studies on the
intrinsic dimension of 2D partition function data sets, by exploring the effect
of volume in 3D Ising data. We find that as opposed to 2D systems for which
this quantity has been successfully used in unsupervised characterizations of
critical phenomena, in the 3D case its estimation is far more challenging. To
circumvent this limitation, we then use the principal component analysis (PCA)
entropy, a "Shannon entropy" of the normalized spectrum of the covariance
matrix. We find a striking qualitative similarity to the thermodynamic entropy,
which the PCA entropy approaches asymptotically. The latter allows us to
extract -- through a conventional finite-size scaling analysis with modest
lattice sizes -- the critical temperature with less than error for both
2D and 3D models while being computationally efficient. The PCA entropy can
readily be applied to characterize correlations and critical phenomena in a
huge variety of many-body problems and suggests a (direct) link between
easy-to-compute quantities and entropies.Comment: Corrected affiliation informatio
Image patch analysis of sunspots and active regions. I. Intrinsic dimension and correlation analysis
The flare-productivity of an active region is observed to be related to its
spatial complexity. Mount Wilson or McIntosh sunspot classifications measure
such complexity but in a categorical way, and may therefore not use all the
information present in the observations. Moreover, such categorical schemes
hinder a systematic study of an active region's evolution for example. We
propose fine-scale quantitative descriptors for an active region's complexity
and relate them to the Mount Wilson classification. We analyze the local
correlation structure within continuum and magnetogram data, as well as the
cross-correlation between continuum and magnetogram data. We compute the
intrinsic dimension, partial correlation, and canonical correlation analysis
(CCA) of image patches of continuum and magnetogram active region images taken
from the SOHO-MDI instrument. We use masks of sunspots derived from continuum
as well as larger masks of magnetic active regions derived from the magnetogram
to analyze separately the core part of an active region from its surrounding
part. We find the relationship between complexity of an active region as
measured by Mount Wilson and the intrinsic dimension of its image patches.
Partial correlation patterns exhibit approximately a third-order Markov
structure. CCA reveals different patterns of correlation between continuum and
magnetogram within the sunspots and in the region surrounding the sunspots.
These results also pave the way for patch-based dictionary learning with a view
towards automatic clustering of active regions.Comment: Accepted for publication in the Journal of Space Weather and Space
Climate (SWSC). 23 pages, 11 figure
Exact Dimensionality Selection for Bayesian PCA
We present a Bayesian model selection approach to estimate the intrinsic
dimensionality of a high-dimensional dataset. To this end, we introduce a novel
formulation of the probabilisitic principal component analysis model based on a
normal-gamma prior distribution. In this context, we exhibit a closed-form
expression of the marginal likelihood which allows to infer an optimal number
of components. We also propose a heuristic based on the expected shape of the
marginal likelihood curve in order to choose the hyperparameters. In
non-asymptotic frameworks, we show on simulated data that this exact
dimensionality selection approach is competitive with both Bayesian and
frequentist state-of-the-art methods
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